Advertisements
Advertisements
प्रश्न
The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is – 30 and the common difference is 8. Find n.
Advertisements
उत्तर
Given that, first term of the first AP(a) = 8
And common difference of the first AP(d) = 20
Let the number of terms in first AP be n.
∵ Sum of first n terms of an AP,
Sn = `n/2[2a + (n - 1)d]`
∴ Sn = `n/2[2 xx 8 + (n - 1)20]`
⇒ Sn = `n/2 (16 + 20n - 20)`
⇒ Sn = `n/2(20n - 4)`
∴ Sn = n(10n – 2) ...(i)
Now, first term of the second AP(a’) = – 30
And common difference of the second AP(d’) = 8
∴ Sum of first 2n terms of second AP,
S2n = `(2n)/2[2a + (2n - 1)d]`
⇒ S2n = n[2(– 30) + (2n – 1)(8)]
⇒ S2n = n[– 60 + 16n – 8)]
⇒ S2n = n[16n – 68] ...(ii)
Now, by given condition,
Sum of first n terms of the first AP = Sum of first 2n terms of the second AP
⇒ Sn = S2n ...[From equations (i) and (ii)]
⇒ n(10n – 2) = n(16n – 68)
⇒ n[(16n – 68) – (10n – 2)] = 0
⇒ n(16n – 68 – 10n + 2) = 0
⇒ n(6n – 66) = 0
⇒ n = 11 ...[∵ n ≠ 0]
Hence, the required value of n is 11.
संबंधित प्रश्न
Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.
Find the sum of first 20 terms of the following A.P. : 1, 4, 7, 10, ........
Ramkali saved Rs 5 in the first week of a year and then increased her weekly saving by Rs 1.75. If in the nth week, her week, her weekly savings become Rs 20.75, find n.
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, .... as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take `pi = 22/7`)
[Hint: Length of successive semicircles is l1, l2, l3, l4, ... with centres at A, B, A, B, ... respectively.]
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceed the second term by 6, find three terms.
Find the sum of the first 13 terms of the A.P: -6, 0, 6, 12,....
Find the sum of first 20 terms of the sequence whose nth term is `a_n = An + B`
Which term of the AP 21, 18, 15, …… is -81?
The first term of an AP is p and its common difference is q. Find its 10th term.
Draw a triangle PQR in which QR = 6 cm, PQ = 5 cm and times the corresponding sides of ΔPQR?
In an A.P. the first term is – 5 and the last term is 45. If the sum of all numbers in the A.P. is 120, then how many terms are there? What is the common difference?
If the sum of a certain number of terms starting from first term of an A.P. is 25, 22, 19, ..., is 116. Find the last term.
Find the sum (−5) + (−8)+ (−11) + ... + (−230) .
In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123. Find n and d, the common differences.
Write the sum of first n odd natural numbers.
Mark the correct alternative in each of the following:
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is
If Sn denote the sum of n terms of an A.P. with first term a and common difference dsuch that \[\frac{Sx}{Skx}\] is independent of x, then
The sum of first n odd natural numbers is ______.
If 18th and 11th term of an A.P. are in the ratio 3 : 2, then its 21st and 5th terms are in the ratio
x is nth term of the given A.P. an = x find x .
The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.
If the third term of an A.P. is 1 and 6th term is – 11, find the sum of its first 32 terms.
If a = 6 and d = 10, then find S10
If the sum of the first m terms of an AP is n and the sum of its n terms is m, then the sum of its (m + n) terms will be ______.
The sum of first 16 terms of the AP: 10, 6, 2,... is ______.
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she travelled carrying a flag?
Calculate the sum of 35 terms in an AP, whose fourth term is 16 and ninth term is 31.
